Biomedical Engineering Reference
In-Depth Information
aries. Coping with the topological changes is another limitation of the deformable
models.
The level set-based segmentation techniques overcome the limits of the clas-
sical deformable models [31, 13, 6, 30]. These techniques handle efficiently topo-
logical changes (curve breaking andmerging) during the course of curve evolution.
In addition, the initial curve needs not be close to the desired solution, and its ini-
tialization can be either manual or automatic. However, since the level set models
evolve using gradient descent, a proper initialization is required for accurate seg-
mentation. The chosen initialization needs an accurate estimate of the parameters
for each class. The Stochastic Expectation Maximization (SEM) algorithm is used
to give initial estimates of class parameters. During the level sets evolution, these
parameters are iteratively re-estimated in order to obtain more accurate segmenta-
tion. Our work differs from that in [7] by its suitability for multimodal images and
due to adaptive estimation of the probability density functions. The segmentation
model as a partial differential equation will be given in detail in the following
section.
Curve/surface evolution has two main representations: explicit and implicit
forms. For each form a partial differential equation is derived to control the
evolution. The details of these approaches are as follows.
4.2.2. Curve and surface evolution theory
In this section we present a brief overview of curve evolution theory. For
a more comprehensive presentation, the interested reader can refer to [32, 33],
among others.
For the sake of simplicity, we formulate the equations of motion of planar
propagating curves. The extension to surfaces in 3D is straightforward. Let
C
(
p, t
):
S
1
×
[0
,
T
)
→
R
2
denote a family of planar closed curves, where
S
1
denotes the unit circle,
t
parameterizes the family, and
p
parameterizes the
curve. Assume that the motion of this family of curves obeys the following partial
differential equation (PDE):
∂
∂t
=
α
·
T
·
N,
and
+
β
C
(
p,
0) =
C
0
(
p
)
,
(1)
where
α
=
α
(
p, t
) and
β
=
β
(
p, t
) denote the tangential and normal velocities,
respectively;
T
=
T
(
p, t
) is the unit tangent vector, and
N
=
N
(
p, t
) is the unit
inward normal vector.
Following the result of Epstein andGage [34], if the normal velocity
β
does not
depend on the parametrization (we say in this case that
β
is a geometric intrinsic
characteristic of the curve), then the solution to equation (2) is identical to the
solution of the following equation of motion:
∂
∂t
=
β
·
N,
and
C
(
p,
0) =
C
0
(
p
)
.
(2)
